Resolve implicit region into explicit one

Does Mathematica provide a way to take, for instance:

ImplicitRegion[x^2 + y^2 <= 1, {x,y}]

and convert it into a region defined explicitly in terms of x and y:

-1 <= x <= 1 && -Sqrt[1-x^2] <= y <= Sqrt[1-x^2]

--

I found Resolve but that only seems to be able to project onto a plane or line.

• Does Reduce@RegionMember[ImplicitRegion[x^2 + y^2 <= 1, {x, y}], {x, y}] meet your needs? – Simon Woods Nov 22 '16 at 22:57
• See CylindricalDecomposition . The example in the doc is exactly what you ask (!) – andre314 Nov 22 '16 at 23:09
• I suspect the applications are far more general than the specific disk illustrated in the question, and thus CylindricalDecomposition will have very limited use. – David G. Stork Nov 22 '16 at 23:16
• @andre Answer? Even the usage message of CylindricalDecomposition seems to describe exactly the OP's problem. "convert it into a region defined explicitly in terms of x and y:", "a decomposition of the region represented by the inequalities ineqs into cylindrical parts whose directions correspond to the successive Subscript[x, i]." – Szabolcs Nov 23 '16 at 11:36
• If someone wants to study a eventual CylindricalDecomposition solution, and give an answer, there's no problem. I don't have time at the moment. – andre314 Nov 23 '16 at 12:50